Base change for intertwiner spaces

Choose $F$-bases of $V$ and $W$, and put $H={\mathrm{Hom}}_F(V,W)$. For $g\in G$, let $T_g:H\to H$ send $f$ to $f{\rho }_V(g)-{\rho }_W(g)f$. By [L2], ${\mathrm{Hom}}_G(V,W)={\bigcap }_{g\in G}\ker T_g$. Since $H$ is finite-dimensional, so is ${\mathrm{End}}_F(H)$. Therefore finitely many $T_{g_1},\dots ,T_{g_r}$ span the subspace generated by all the $T_g$, and ${\mathrm{Hom}}_G(V,W)={\bigcap }_{i=1}^r\ker T_{g_i}$.

The chosen bases identify $E{\otimes }_{F}H$ with ${\mathrm{Hom}}_E(E{\otimes }_{F}V,E{\otimes }_{F}W)$ and identify each $E{\otimes }_F{T}_g$ with the corresponding intertwining equation over $E$. Gaussian elimination over $F$ gives a basis of the finite simultaneous kernel in step 1.1 and extends it to a basis of $H$; the same row operations remain valid over $E$. Hence scalar extension preserves this finite kernel. Every $T_g$ is an $F$-linear combination of the selected $T_{g_i}$, so the simultaneous kernel over $E$ for all $g$ equals that finite kernel and, by [L2], is ${\mathrm{Hom}}_G(E{\otimes }_{F}V,E{\otimes }_{F}W)$.

The resulting identification sends $a\otimes f$ to the displayed map, which agrees with [L1]. Therefore it is the asserted $E$-linear isomorphism.